Polygonal Knot Research Articles

Knotting spectrum of polygonal knots in extreme confinement

Random knot models are often used to measure the types of entanglements one would expect to observe in an unbiased system with some given physical property or set of properties. In nature, macromolecular chains often exist in extreme confinement. Current techniques for sampling random polygons in confinement are limited. In this paper, we gain insight into the knotting behavior of random polygons in extreme confinement by studying random polygons restricted to cylinders, where each edge connects the top and bottom disks of the cylinder. The knot spectrum generated by this model is compared to the knot spectrum of rooted equilateral random polygons in spherical confinement. Due to the rooting, such polygons require a radius of confinement R ⩾ 1. We present numerical evidence that the polygons generated by this simple cylindrical model generate knot probabilities that are equivalent to spherical confinement at a radius of R ≈ 0.62. We then show how knot complexity and the relative probability of different classes of knot types change as the length of the polygon increases in the cylindrical polygons.

The limit shapes of midpoint polygons in ℝ3

For a polygon in the [Formula: see text]-dimensional Euclidean space, we give two kinds of normalizations of its [Formula: see text]th midpoint polygon by a homothetic transformation and an affine transformation, respectively. As [Formula: see text] goes to infinity, the normalizations will approach “regular” polygons inscribed in an ellipse and a generalized Lissajous curve, respectively, where the curves may be degenerate. The most interesting case is when [Formula: see text], where polygons with all its [Formula: see text]th midpoint polygons knotted are discovered and discussed. Such polygonal knots can be seen as a discrete version of the Lissajous knots.

Formalizing polygonal knot origami

We present computer-assisted construction of regular polygonal knots by origami. The construction is completed with an automated proof based on algebraic methods. Given a rectangular origami or a finite tape, of an adequate length, we can construct the simplest knot by three folds. The shape of the knot is made to be a regular pentagon if we fasten the knot tightly without distorting the tape. We perform the analysis of the knot fold further formally towards the automated construction and verification. In particular, we show the construction and proof of regular pentagonal and heptagonal knots. We employ a software tool called Eos (e-origami system), which incorporates the extension of Huzita's basic fold operations for construction, and Gröbner basis computation for proving. Our study yields more mathematical rigor and in-depth results about the polygonal knots.

High resolution portrait of the ideal trefoil knot

The shape of the tightest polygonal trefoil knot with N = 200 640 vertices, constructed by the authors using an appropriately modified finite element method, is carefully analyzed. The large number of vertices provide a spatial structure with very precise plots of its curvature and torsion. Based on the increased accuracy of the data, the authors are able to formulate new conjectures concerning the key features of the shape of the ideal trefoil knot.

Equilateral stick number of knots

An equilateral stick number s=(K) of a knot K is defined to be the minimal number of sticks required to construct a polygonal knot of K which consists of equal length sticks. Rawdon and Scharein [Upper bounds for equilateral stick numbers, in Physical Knots: Knotting, Linking, and Folding Geometric Objects in ℝ3, Contemporary Mathematics, Vol. 304 (American Mathematical Society, Providence, RI, 2002), pp. 55–76] found upper bounds for the equilateral stick numbers of all prime knots through 10 crossings by using algorithms in the software KnotPlot. In this paper, we find an upper bound on the equilateral stick number of a non-trivial knot K in terms of the minimal crossing number c(K) which is s=(K) ≤ 2c(K) + 2. Moreover if K is a non-alternating prime knot, then s=(K) ≤ 2c(K) - 2. Furthermore we find another upper bound on the equilateral stick number for composite knots which is s=(K1♯K2) ≤ 2c(K1) + 2c(K2).

The Generation of Random Equilateral Polygons

Freely jointed random equilateral polygons serve as a common model for polymer rings, reflecting their statistical properties under theta conditions. To generate equilateral polygons, researchers employ many procedures that have been proved, or at least are believed, to be random with respect to the natural measure on the space of polygonal knots. As a result, the random selection of equilateral polygons, as well as the statistical robustness of this selection, is of particular interest. In this research, we study the key features of four popular methods: the Polygonal Folding, the Crankshaft Rotation, the Hedgehog, and the Triangle Methods. In particular, we compare the implementation and efficacy of these procedures, especially in regards to the population distribution of polygons in the space of polygonal knots, the distribution of edge vectors, the local curvature, and the local torsion. In addition, we give a rigorous proof that the Crankshaft Rotation Method is ergodic.

MÖBIUS TRANSFORMATIONS OF POLYGONS AND PARTITIONS OF 3-SPACE

The image of a polygonal knot K under a spherical inversion of ℝ3 ∪ ∞ is a simple closed curve made of arcs of circles, perhaps some line segments, having the same knot type as the mirror image of K. But suppose we reconnect the vertices of the inverted polygon with straight lines, making a new polygon [Formula: see text]. This may be a different knot type. For example, a certain 7-segment figure-eight knot can be transformed to a figure-eight knot, a trefoil, or an unknot, by selecting different inverting spheres. Which knot types can be obtained from a given original polygon K under this process? We show that for large n, most n-segment knot types cannot be reached from one initial n-segment polygon, using a single inversion or even the whole Möbius group. The number of knot types is bounded by the number of complementary domains of a certain system of round 2-spheres in ℝ3. We show the number of domains is at most polynomial in the number of spheres, and the number of spheres is itself a polynomial function of the number of edges of the original polygon. In the analysis, we obtain an exact formula for the number of complementary domains of any collection of round 2-spheres in ℝ3. On the other hand, the number of knot types that can be represented by n-segment polygons is exponential in n. Our construction can be interpreted as a particular instance of building polygonal knots in non-Euclidean metrics. In particular, start with a list of n vertices in ℝ3 and connect them with arcs of circles instead of line segments: Which knots can be obtained? Our polygonal inversion construction is equivalent to picking one fixed point p ∈ ℝ3 and replacing each edge of K by an arc of the circle determined by p and the endpoints of the edge.

POLYGONAL KNOT SPACE NEAR ROPELENGTH-MINIMIZED KNOTS

For a polygonal knot K, it is shown that a tube of radius R(K), the polygonal thickness radius, is an embedded torus. Given a thick configuration K, perturbations of size r < R(K) define satellite structures, or local knotting. We explore knotting within these tubes both theoretically and numerically. We provide bounds on perturbation radii for which one can obtain small trefoil and figure-eight summands and use Monte Carlo simulations to estimate the relative probabilities of these structures as a function of the number of edges.

POLYGONAL APPROXIMATION AND ENERGY OF SMOOTH KNOTS

We establish a fundamental connection between smooth and polygonal knot energies, showing that the Minimum Distance Energy for polygons inscribed in a smooth knot converges to the Möbius Energy of the smooth knot as the polygons converge to the smooth knot. For this to work, the polygons must converge in a "nice" way, and the energies must be correctly regularized. We determine an explicit error bound for the convergence.

On stick number of knots and links

Abstract The DNA strand is made up of small rigid sticks of sugar, phosphorus, nucleotide proteins and hydrogen bond. A cyclic molecule can be modelled by a sequence of sticks glued end-to-end so that the last one is also glued to the first. Thus we have mathematical stick knots or polygonal knots. This gives rise to some questions: For instance, what is the smallest number of atoms needed to construct a nontrivial knotted molecule? For such stick numbers, Negami gave an upper and lower bound. In the case when a given knot reaches its lower bound in Negami’s inequality, we gave a specific relation between both of its stick and crossing numbers. Hence we conclude that not one of the knots, at least, to 26 crossings can achieve its lower bound. In other words every knotted molecule of atoms up to 26 will have a number of bonds more than the lower bound in Negami’s inequality. Subsequently it is shown that some interesting class of torus knots does not achieve that lower bound in Negami’s inequality.

Computing the Writhing Number of a Polygonal Knot

The writhing number measures the global geometry of a closed space curve or knot. We show that this measure is related to the average winding number of its Gauss map. Using this relationship, we give an algorithm for computing the writhing number for a polygonal knot with n edges in time roughly proportional to n1.6. We also implement a different, simple algorithm and provide experimental evidence for its practical efficiency.

Can Computers Discover Ideal Knots?

We discuss the relationship between polygonal knot energies and smooth knot energies, concentrating on ropelength. We show that a smooth knot can be inscribed in a polygonal knot in such a way that the ropelength values are close. For a given knot type, we show that polygonal ropelength minima exist and that the minimal polygonal ropelengths converge to the minimal ropelength of the smooth knot type. A subsequence of these polygons converges to a smooth ropelength minimum. Thus, ropelength minimizations performed on polygonal knots do, in fact, approximate ropelength minimizations for smooth knots.

Normal holonomy and writhing number of polygonal knots

The normal holonomy of a polygonal knot is a geometrical invariant which is closely related to the writhing number. We show that normal holonomy fibers the space of knots over the circle and deduce that the writhing number fibers the space of knots over the real line. Consequently, two isotopic knots which have the same writhing number are isotopic through a family of knots having the same writhing number. In a similar vein, two isotopic knots having zero holonomy are isotopic through a family of such knots if and only if they have the same autoparallel linking number. More generally, the definition of normal holonomy makes sense for immersed polygonal knots. This time normal holonomy fibers the space of immersed knots over the circle, but now there are only two isotopy classes of immersed knots of zero holonomy.

Teoría de nudos geométricos e isotopía poligonal

El espacio de los polígonos de n lados, inmersos en el espacio euclídeo de tres dimensiones, consiste de una variedad suave en la cual los puntos corresponden a nudos lineales a trozos o “geométricos”, mientras que los arcos corresponden a isotopías que preservan la estructura geométrica de esos nudos. Se describe la topología de estos espacios para los casos n = 6 y n = 7. En ambos casos, cada espacio consta de cinco componentes, aunque contiene sólo tres (cuando n = 6) o cuatro (cuando n = 7) tipos topológicos de nudos. Por lo tanto la “equivalencia geométrica de nudos” es estrictamente más fuerte que la equivalencia topológica. Este hecho se demuestra con el nudo trébol hexagonal y el nudo doble heptagonal, los cuales, a diferencia de sus contrapartes topológicas, no son reversibles. Se discutirán también las extensiones de estos resultados a los casos n ≥ 8.

THE MINIMUM OF KNOT ENERGY FUNCTIONS

In this paper we discuss some fundamental issues regarding knot energy functions. These include the existence of minimum values of energy functions of smooth knots and energy functions of polygonal knots within a knot type, the convergence of these minimum values in the case of polygonal knot energy and the convergence of the corresponding polygons where these minimum values are attained. When the polygonal knot energy is derived from a smooth knot energy, will the minimal polygonal knot energies converge to the infimum of the smooth knot energy? Do the corresponding polygons converge to a smooth knot at which the smooth energy achieves its minimal value? We show that one cannot expect these to be true in general and outline certain conditions that would ensure a positive answer to some of the above questions.

Digital knots

We define digital (6-)knots in Z 3, and discuss how they are related to isothetic polygonal knots in R 3. We introduce a class of topology-preserving “simple deformations” (SD) of digital images, and show that isomorphic knots have digitizations that differ by SD, and conversely. We also define “regular position” for isothetic links and digital links, and show that SD can be used to put any digital knot into such a position and to perform “Reidemeister moves” on the resulting projection.

The computational complexity of knot and link problems

We consider the problem of deciding whether a polygonal knot in 3-dimensional Euclidean space is unknotted, ie., capable of being continuously deformed without self-intersection so that it lies in a plane. We show that this problem, UNKNOTTING PROBLEM is in NP. We also consider the problem, SPLITTING PROBLEM of determining whether two or more such polygons can be split, or continuously deformed without self-intersection so that they occupy both sides of a plane without intersecting it. We show that it also is in NP. Finally, we show that the problem of determining the genus of a polygonal knot (a generalization of the problem of determining whether it is unknotted) is in PSPACE. We also give exponential worst-case running time bounds for deterministic algorithms to solve each of these problems. These algorithms are based on the use of normal surfaces and decision procedures due to W. Haken, with recent extensions by W. Jaco and J. L. Tollefson.

The complexity of lattice knots

A family of polygonal knots K n on the cubical lattice is constructed with the property that the quotient of length L( K n ) over the crossing number Cr( K n ) approaches zero as L approaches infinity. More precisely Cr(K n) = O(L(K n) 4 3 ) . It is shown that this construction is optimal in the sense that for any knot K on the cubical lattice with length L and Cr crossings Cr ⩽ 3.2L 4 3 .

In Search of a Good Polygonal Knot Energy

An energy function on knots is a scale-invariant function from knot conformations into non-negative real numbers. The infimum of an energy function is an invariant which defines "canonical conformation(s)" of a knot in three space. These are not necessarily unique, and, in some cases, may even be singular. Many hierarchies of energy functions for knots in the mathematical and physical science literature have been studied, each energy function with its own characteristic set of properties. In this paper we focus on the energy functions of equilateral polygonal knots. These energy functions are important in computer studies of knot energies, and are often defined as discrete versions of energy functions defined on smooth knots. Energy functions on equilateral polygonal knots turn out to be ill-behaved in many cases. To characterize a "good" polygonal knot energy we introduce the concepts of asymptotically finite and asymptotically smooth energy functions of equilateral polygonal knots. Energy functions which are both asymptotically finite and smooth tend to have food limiting behavior (as the number of edges goes to infinity). We introduce a new energy function of equilateral polygonal knots, and show that it is both asymptotically finite and smooth. In addition, we compute this energy for several knots using simulated annealing.

AN ELEMENTARY INVARIANT OF KNOTS

We indicate how certain invariants can be computed for polygonal knots with few edges. This leads to conclusions regarding the minimal number of edges required to represent certain knots.